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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fréchet derivative ： ウィキペディア英語版 Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on Banach spaces. The Fréchet derivative should be contrasted to the more general Gâteaux derivative which is a generalization of the classical directional derivative. The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis. ==Definition== Let ''V'' and ''W'' be Banach spaces, and $U\backslash subset\; V$ be an open subset of ''V''. A function ''f'' : ''U'' → ''W'' is called ''Fréchet differentiable'' at $x\; \backslash in\; U$ if there exists a bounded linear operator $A:V\backslash to\; W$ such that :$\backslash lim\_\; \backslash frac\; =\; 0.$ The limit here is meant in the usual sense of a limit of a function defined on a metric space (see Functions on metric spaces), using ''V'' and ''W'' as the two metric spaces, and the above expression as the function of argument ''h'' in ''V''. As a consequence, it must exist for all sequences $\backslash langle\; h\_n\backslash rangle\_^$ of non-zero elements of ''V'' which converge to the zero vector $h\_n0.$ Equivalently, the first-order expansion holds, in Landau notation :$f(x\; +\; h)\; =\; f(x)\; +\; Ah\; +o(h).$ If there exists such an operator ''A'', it is unique, so we write $Df(x)=A$ and call it the (Fréchet) derivative of ''f'' at ''x''. A function ''f'' that is Fréchet differentiable for any point of ''U'' is said to be C1 if the function :$Df:U\backslash to\; B(V,W)\; ;\; x\; \backslash mapsto\; Df(x)$ is continuous. Note that this is not the same as requiring that the map $Df(x)\; :\; V\; \backslash to\; W$ be continuous for each value of $x$ (which is assumed). This notion of derivative is a generalization of the ordinary derivative of a function on the real numbers ''f'' : R → R since the linear maps from R to R are just multiplication by a real number. In this case, ''Df''(''x'') is the function $t\; \backslash mapsto\; tf\text{'}(x)$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fréchet derivative」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.